translations do not preserve the inner product. I am not sure exactly what you're asking. Are you asking whether rigid transformations preserve some structure in addition to the inner product (and whatever comes from it)?
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Qiaochu YuanSep 21 '10 at 14:25

Sorry. Made a mistake. Now making changes to the original post.
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TimSep 21 '10 at 15:12

1 Answer
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If by "rotation" you mean "orientation-preserving isometry of $\mathbb{R}^n$ fixing the origin," then

the spaces between which rigid transformations are morphisms are the "affine oriented (finite-dimensional) inner product spaces" over $\mathbb{R}$, by which I mean torsors over a finite-dimensional oriented inner product space over $\mathbb{R}$,

the spaces between which rotations are morphisms are the oriented finite-dimensional inner product spaces over $\mathbb{R}$, and

the spaces between which projective transformations are morphisms are the projective spaces over $\mathbb{R}$.

I cannot off the top of my head think of a good name for the spaces between which similarities are morphisms. I think you are looking for "conformal affine spaces," e.g. torsors over $\mathbb{R}^n$ equipped with a notion of oriented angle (but not the full inner product).

The truth of yur statement of course depends on what you mean by structure :) Is everything preserved by motions definable in terms of/determined by the metric and the orientation?
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Mariano Suárez-Alvarez♦Sep 21 '10 at 14:36